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Stochastic logarithmic Schrödinger equations driven by Lévy noise

Jiahui Zhu, Jianliang Zhai

TL;DR

The paper studies the stochastic logarithmic Schrödinger equation with saturated multiplicative Lévy noise and proves global well-posedness in the energy space $W=H^1(\mathbb{R}^d)\cap V$ by regularizing the logarithmic nonlinearity through $L_\varepsilon(u)=\log\left(\frac{|u|+\varepsilon}{1+\varepsilon|u|}\right)$ and solving the regularized problem using the Marcus canonical integral. A key feature is constructing solutions as strong limits of the approximations, yielding strong convergence in $L^2_{\text{loc}}$ without relying on weak compactness. The analysis relies on uniform $H^1$-type and Orlicz (entropy) estimates, mass conservation, and the strong convergence of the nonlinear term $u_\varepsilon L_\varepsilon(u_\varepsilon)$ to $u\log|u|$. The methods accommodate arbitrary sign of the logarithmic coefficient $\lambda$ and exploit the Marcus calculus to handle jump noise, providing a robust framework for well-posedness in the presence of discontinuous stochastic forcing. The results extend deterministic and Gaussian-noise theories to jump-driven settings, with potential implications for nonlinear wave propagation in saturable media under random jumps.

Abstract

In this paper, we study the stochastic logrithmic Schrödinger equation with saturated nonlinear multiplicative Lévy noise. The global well-posedness is established for the stochastic logrithmic Schrödinger equation in an appropriate Orlicz space by construct solutions of a regularized equation converging strongly to a solution to the original equation.

Stochastic logarithmic Schrödinger equations driven by Lévy noise

TL;DR

The paper studies the stochastic logarithmic Schrödinger equation with saturated multiplicative Lévy noise and proves global well-posedness in the energy space by regularizing the logarithmic nonlinearity through and solving the regularized problem using the Marcus canonical integral. A key feature is constructing solutions as strong limits of the approximations, yielding strong convergence in without relying on weak compactness. The analysis relies on uniform -type and Orlicz (entropy) estimates, mass conservation, and the strong convergence of the nonlinear term to . The methods accommodate arbitrary sign of the logarithmic coefficient and exploit the Marcus calculus to handle jump noise, providing a robust framework for well-posedness in the presence of discontinuous stochastic forcing. The results extend deterministic and Gaussian-noise theories to jump-driven settings, with potential implications for nonlinear wave propagation in saturable media under random jumps.

Abstract

In this paper, we study the stochastic logrithmic Schrödinger equation with saturated nonlinear multiplicative Lévy noise. The global well-posedness is established for the stochastic logrithmic Schrödinger equation in an appropriate Orlicz space by construct solutions of a regularized equation converging strongly to a solution to the original equation.

Paper Structure

This paper contains 5 sections, 10 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Approximating equation
  4. Uniform estimates
  5. Proof of Theorem \ref{['Them-main']}

Key Result

Theorem 1.1

Suppose that Assumption assum-main holds. Let $\lambda \in \mathbb{R} \backslash\{0\}$ and $p\geqslant 2$. Let $0<T<\infty$. If $u_0\in L^p(\Omega,\mathcal{F}_0;W)$, then there exists a unique global solution $u\in L^p(\Omega;L^{\infty}(0,T;W))$ to equation SLogSE in the sense of Definition defi-sol and the mass conservation holds $\mathbb{P}$-a.s. for all $t\in[0,T]$,

Theorems & Definitions (23)

  • Theorem 1.1
  • Lemma 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 3.1
  • Lemma 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • ...and 13 more